Travelling wave profiles in some models with nonlinear diffusion

We study some properties of the monotone solutions of the boundary value problem(P(u′))′-cu′+f(u)=0,u(-∞)=0,u(+∞)=1,where f is a continuous function, positive in (0,1) and taking the value zero at 0 and 1, and P may be an increasing homeomorphism of [0,1) or [0,+∞) onto [0,+∞). This problem arises when we look for travelling waves for the reaction diffusion equation∂u∂t=∂∂xP∂u∂x+f(u)with the parameter c representing the wave speed. A possible model for the nonlinear diffusion is the relativistic curvature operator P(v)=v1-v2. The same ideas apply when P is given by the one-dimensional p-Laplacian P(v)=vp-2v. In this case, an advection term is also considered. We show that, as for the classical Fisher–Kolmogorov–Petrovski–Piskounov equations, there is an interval of admissible speeds [c∗,+∞) and we give characterisations of the critical speed c∗. We also present some examples of exact solutions.